## Note on Vector

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### Product of the vectors

Owing to the different way in which vectors occur in various physical problems, the product of two vectors $$\overrightarrow {a}$$ and $$\overrightarrow {b}$$ is defined in the following two ways:

1. scalar product or dot product $$\overrightarrow a$$ . $$\overrightarrow b$$ (read as $$\overrightarrow a$$ dot $$\overrightarrow b$$)
2. vector product or cross product $$\overrightarrow a$$× $$\overrightarrow b$$ (read as $$\overrightarrow a$$ cross $$\overrightarrow b$$).

The dot product $$\overrightarrow a$$ . $$\overrightarrow b$$ gives a scalar result while the cross product $$\overrightarrow a$$× $$\overrightarrow b$$ gives a vector result.

#### Scalar product of two vectors

The scalar product of two vectors $$\overrightarrow a$$ and $$\overrightarrow b$$ is defined as the product of the magnitude of two vectors multiplied by the cosine of the angle $$\theta$$ between their directions.

Thus, $$\overrightarrow a$$ . $$\overrightarrow b$$ = |$$\overrightarrow a$$| |$$\overrightarrow b$$| cos$$\theta$$ = ab cos$$\theta$$

where, |$$\overrightarrow a$$| = a and |$$\overrightarrow b$$| = b.

Now,

Draw perpendicular BM from B to OA.

Here,

$$\overrightarrow {OA}$$ = $$\overrightarrow a$$ and $$\overrightarrow {OB}$$ = $$\overrightarrow b$$

Now,

\begin{align*} \overrightarrow a . \overrightarrow b &= |\overrightarrow a| |\overrightarrow b| cos \theta\\ &= a b cos\theta\\ &= (OA) (OB) cos\theta\\ &= OA (OB cos\theta)\\ &= (OA)(OM)\\ &= (magnitude\;of\;\overrightarrow a) (component\;of\;\overrightarrow b\;in\;the\;direction\;of\;\overrightarrow a)\\ \end{align*}

So, it is clear that the scalar product of two vectors is equivalent to the product of the magnitude of one vector with the component of the other vector in the direction of this vector.

If we write $$\overrightarrow a$$ . $$\overrightarrow b$$, the rotation of $$\overrightarrow a$$ towards $$\overrightarrow b$$ is anticlockwise and the angle $$\theta$$ is taken to be positive.

∴ $$\overrightarrow a$$ . $$\overrightarrow b$$ = ab cos$$\theta$$

If we write $$\overrightarrow b$$ . $$\overrightarrow a$$, the rotation of $$\overrightarrow b$$ towards $$\overrightarrow a$$ is clockwise and the angle $$\theta$$ is taken to be negative.

∴ $$\overrightarrow b$$ . $$\overrightarrow a$$ = b a cos(-$$\theta$$) = ba cos$$\theta$$

Hence, $$\overrightarrow a$$ . $$\overrightarrow b$$ = $$\overrightarrow b$$ . $$\overrightarrow a$$

Thus, scalar product is commulative.

Let us consider two points A(a1, a2) and B(b1, b2) in the plane. Then,

position vector of A = $$\overrightarrow {OA}$$ = $$\overrightarrow a$$ = $$\begin{pmatrix} a_1\\ a_2\\ \end{pmatrix}$$

position vector of B = $$\overrightarrow {OB}$$ = $$\overrightarrow b$$ = $$\begin{pmatrix} b_1\\ b_2\\ \end{pmatrix}$$

Magnitudes of $$\overrightarrow a$$ and $$\overrightarrow b$$ are

|$$\overrightarrow {OA}$$| = OA = a = |$$\overrightarrow a$$|

|$$\overrightarrow {OB}$$| = OB = b = |$$\overrightarrow b$$|

Let $$\angle$$XOA =β, $$\angle$$XOB =α and $$\angle$$AOB =θ. Then,α -β =θ.

Draw perpendiculars AM and BN from A and B to the x-axis. Then,

OM = a1, MA = a2, ON = b1 and NB = b2.

From the right-angled triangle OMA,

cosβ = $$\frac {OM}{OA}$$ = $$\frac {a_1}a$$∴ a1 = a cosβ

sinβ = $$\frac {MA}{OA}$$ = $$\frac {a_2}a$$∴ a2 = a sinβ

Similarly,

From the right-angled triangle ONB,

b1 = b cosα and b2 = b sinα

Now,

\begin{align*} a_1b_1 + a_2b_2 &= a cosβ b cosα + a sinβ sinα\\ &= ab cos (α - β)\\ &= |\overrightarrow a| |\overrightarrow b| cos \theta...................(i)\\ \end{align*}

But,

By the deefinition of scalar product of two vectors $$\overrightarrow a$$ and $$\overrightarrow b$$,

Now,

From (i) and (ii),

$$\overrightarrow a$$ . $$\overrightarrow b$$ = a1b1 + a2b2.

This result leads us to define the scalar product of two vectors in another way.

Let $$\overrightarrow a$$ = $$\begin{pmatrix} a_1\\ a_2\\ \end{pmatrix}$$ and $$\overrightarrow b$$ = $$\begin{pmatrix} b_1\\ b_2\\ \end{pmatrix}$$ be two vectors. Then the scalar product of $$\overrightarrow a$$ and $$\overrightarrow b$$ is denoted by $$\overrightarrow a$$ . $$\overrightarrow b$$ and is defined by $$\overrightarrow a$$ . $$\overrightarrow b$$ = $$\begin{pmatrix} a_1\\ a_2\\ \end{pmatrix}$$ and $$\overrightarrow b$$ .$$\begin{pmatrix} b_1\\ b_2\\ \end{pmatrix}$$ = a1b1 + a2b2.

Again,

From (i),

|$$\overrightarrow a$$| |$$\overrightarrow b$$| cos$$\theta$$ = a1b1 + a2b2

or, cos$$\theta$$ = $$\frac {a_1b_1 + a_2b_2}{|\overrightarrow a| |\overrightarrow b|}$$

This result gives us angle between two vectors $$\overrightarrow a$$ and $$\overrightarrow b$$.

$$\theta$$ = cos-1 $$\frac {a_1b_1 + a_2b_2}{|\overrightarrow a| |\overrightarrow b|}$$

##### Properties of Scalar Product

The following properties are satisfied by the scalar product of vectors:

Let $$\overrightarrow a$$, $$\overrightarrow b$$ and $$\overrightarrow c$$ be three vectors.

1. Commulative Property: $$\overrightarrow a$$ . $$\overrightarrow b$$ = $$\overrightarrow b$$ . $$\overrightarrow a$$
2. Distributive Property: $$\overrightarrow a$$ . ($$\overrightarrow b$$ + $$\overrightarrow c$$) = $$\overrightarrow a$$ . $$\overrightarrow b$$ + $$\overrightarrow a$$ . $$\overrightarrow c$$
3. Associative Property:m $$\overrightarrow a$$ . n $$\overrightarrow b$$ = mn ($$\overrightarrow a$$ . $$\overrightarrow b$$)

#### Perpendicular Vectors

Let $$\overrightarrow a$$ = $$\begin{pmatrix} a_1\\ a_2\\ \end{pmatrix}$$ and $$\overrightarrow b$$ = $$\begin{pmatrix} b_1\\ b_2\\ \end{pmatrix}$$ be two vectors. If $$\overrightarrow a$$ and $$\overrightarrow b$$ are perpendicular to each other, then the angle between $$\overrightarrow a$$ and $$\overrightarrow b$$ is $$\theta$$ =- 90°.

Now,

$$\overrightarrow a$$ . $$\overrightarrow b$$ = |$$\overrightarrow a$$| |$$\overrightarrow b$$| cos$$\theta$$ = ab cos 90° = 0

Conversely,

Let $$\overrightarrow a$$ . $$\overrightarrow b$$ = 0

Then,

|$$\overrightarrow a$$| |$$\overrightarrow b$$| cos$$\theta$$ = 0

or, ab cos$$\theta$$ = 0

or, cos$$\theta$$ = 0

∴$$\theta$$ = 90°

Thus, if two vectors are perpendicular to each other (or orthogonal), their scalar product is zero.

#### Parallel Vector

Let $$\overrightarrow a$$ and $$\overrightarrow b$$ be two vectors. If $$\overrightarrow a$$ and $$\overrightarrow b$$ are parallel to each other then the angle between them is 0° or 180°.

Now,

If $$\theta$$ = 0°, $$\overrightarrow a$$ . $$\overrightarrow b$$ = |$$\overrightarrow a$$| |$$\overrightarrow b$$| cos$$\theta$$ = ab cos 0° = ab

If $$\theta$$ = 180°, $$\overrightarrow a$$ . $$\overrightarrow b$$ = |$$\overrightarrow a$$| |$$\overrightarrow b$$| cos$$\theta$$ = ab cos 180° = -ab

Thus, two vector $$\overrightarrow a$$ and $$\overrightarrow b$$ are parallel to each other if $$\overrightarrow a$$ . $$\overrightarrow b$$ = ab or, $$\overrightarrow a$$ . $$\overrightarrow b$$ = -ab.

#### Length of a vector (Modulus of a vector)

Let $$\overrightarrow a$$ = $$\begin{pmatrix} a_1\\ a_2\\ \end{pmatrix}$$ be a plane vector.

Then, $$\overrightarrow a$$ . $$\overrightarrow a$$ = $$\begin{pmatrix} a_1\\ a_2\\ \end{pmatrix}$$ . $$\begin{pmatrix} a_1\\ a_2\\ \end{pmatrix}$$ = a1a1 + a2a2= a12 + a22 = a2

∴ a = $$\sqrt {\overrightarrow a. \overrightarrow a}$$

Hence, the length of a vector $$\overrightarrow a$$ is the positive square root of the scalar product $$\overrightarrow a$$ . $$\overrightarrow a$$.

The scalar product of a vector with itself is often written as the square of the vector.

So, $$\overrightarrow a$$ . $$\overrightarrow a$$ = a2 i.e. $$\overrightarrow {a^2}$$ = a2

#### Some simple identities

1. ($$\overrightarrow a$$ + $$\overrightarrow b$$) = $$\overrightarrow {a^2}$$ + 2$$\overrightarrow a$$ . $$\overrightarrow b$$ + $$\overrightarrow {b}$$ = a2 + 2 $$\overrightarrow a$$ . $$\overrightarrow b$$ + b2
($$\overrightarrow a$$ + $$\overrightarrow b$$) =($$\overrightarrow a$$ + $$\overrightarrow b$$) .($$\overrightarrow a$$ + $$\overrightarrow b$$) = $$\overrightarrow a$$ . $$\overrightarrow a$$ + $$\overrightarrow a$$ . $$\overrightarrow b$$ + $$\overrightarrow b$$ . $$\overrightarrow a$$ + $$\overrightarrow b$$ . $$\overrightarrow b$$ = $$\overrightarrow a$$ + 2$$\overrightarrow a$$ . $$\overrightarrow b$$ + $$\overrightarrow {b^2}$$ = a2 + 2$$\overrightarrow a$$ . $$\overrightarrow b$$ + $$\overrightarrow b$$
2. ($$\overrightarrow a$$ - $$\overrightarrow b$$)2 = $$\overrightarrow {a^2}$$ - 2 $$\overrightarrow a$$ . $$\overrightarrow a$$ + $$\overrightarrow {a^2}$$ = a2 - 2$$\overrightarrow a$$ . $$\overrightarrow b$$ + b2
($$\overrightarrow a$$ - $$\overrightarrow b$$)2 =($$\overrightarrow a$$ - $$\overrightarrow b$$) .($$\overrightarrow a$$ - $$\overrightarrow b$$)= $$\overrightarrow a$$ . $$\overrightarrow a$$ - $$\overrightarrow a$$ . $$\overrightarrow b$$ - $$\overrightarrow b$$ . $$\overrightarrow a$$ + $$\overrightarrow b$$ . $$\overrightarrow b$$ = $$\overrightarrow a$$ - 2$$\overrightarrow a$$ . $$\overrightarrow b$$ + $$\overrightarrow b$$ = a2 - 2$$\overrightarrow a$$ . $$\overrightarrow b$$ + b2
3. ($$\overrightarrow a$$ + $$\overrightarrow b$$) .($$\overrightarrow a$$ - $$\overrightarrow b$$ = $$\overrightarrow {a^2}$$ - $$\overrightarrow {b^2}$$ = a2 - b2
($$\overrightarrow a$$ + $$\overrightarrow b$$) .($$\overrightarrow a$$ - $$\overrightarrow b$$ = $$\overrightarrow a$$ . $$\overrightarrow a$$ - $$\overrightarrow a$$ . $$\overrightarrow b$$ + $$\overrightarrow b$$ . $$\overrightarrow b$$ = $$\overrightarrow {a^2}$$ - $$\overrightarrow b$$ = a2 - b2

#### Mutually perpendicular unit vector $$\overrightarrow i$$ and $$\overrightarrow j$$

Let OX and OY be two mutually perpendicular straight lines. Then the unit vector along OX and OY denoted by $$\overrightarrow i$$ and $$\overrightarrow j$$ are defined by $$\overrightarrow i$$ = $$\begin{pmatrix} 1\\ 0\\ \end{pmatrix}$$ and $$\overrightarrow j$$ = $$\begin{pmatrix} 0\\ 1\\ \end{pmatrix}$$.

Now,

$$\overrightarrow i$$ . $$\overrightarrow i$$ = $$\overrightarrow {i^2}$$ =$$\begin{pmatrix} 1\\ 0\\ \end{pmatrix}$$ . $$\begin{pmatrix} 1\\ 0\\ \end{pmatrix}$$ = 1 + 0 = 1

$$\overrightarrow j$$ . $$\overrightarrow j$$ = $$\overrightarrow {j^2}$$ =$$\begin{pmatrix} 0\\ 1\\ \end{pmatrix}$$ . $$\begin{pmatrix} 0\\ 1\\ \end{pmatrix}$$ = 0 + 1 = 1

$$\overrightarrow i$$ . $$\overrightarrow j$$ = $$\begin{pmatrix} 1\\ 0\\ \end{pmatrix}$$ . $$\begin{pmatrix} 0\\ 1\\ \end{pmatrix}$$ = 0 + 0 = 0

$$\overrightarrow j$$ . $$\overrightarrow i$$ = $$\begin{pmatrix} 0\\ 1\\ \end{pmatrix}$$ . $$\begin{pmatrix} 1\\ 0\\ \end{pmatrix}$$ = 0 + 0 = 0

The value of scalar product of $$\overrightarrow i$$ and $$\overrightarrow j$$ can be remembered from the table given alongside.

#### Representation of a vector in terms of unit vectors

Let $$\overrightarrow a$$ = $$\begin{pmatrix}x\\ y\\ \end{pmatrix}$$ be a vector. It can be written as

$$\overrightarrow a$$ = $$\begin{pmatrix}x\\ y\\ \end{pmatrix}$$ = $$\begin{pmatrix} x\\ 0\\ \end{pmatrix}$$ = $$\begin{pmatrix} 0\\ y\\ \end{pmatrix}$$ = x$$\begin{pmatrix}1\\ 0\\ \end{pmatrix}$$ + y$$\begin{pmatrix}0\\ 1\\ \end{pmatrix}$$ = x$$\overrightarrow i$$ + y$$\overrightarrow j$$.

Similarly,

If $$\overrightarrow a$$ = nx + y$$\overrightarrow j$$, then:

$$\overrightarrow a$$= x$$\overrightarrow i$$ + y$$\overrightarrow j$$ = x$$\begin{pmatrix}1\\ 0\\ \end{pmatrix}$$ + y$$\begin{pmatrix}0\\ 1\\ \end{pmatrix}$$ = $$\begin{pmatrix}x\\ 0\\ \end{pmatrix}$$ + $$\begin{pmatrix}0\\ y\\ \end{pmatrix}$$ = $$\begin{pmatrix}x\\ y\\ \end{pmatrix}$$

Hence, every plane vector $$\begin{pmatrix}x\\ y\\ \end{pmatrix}$$ can be represented by x$$\overrightarrow i$$ + y$$\overrightarrow j$$ and conversely.

#### Vector operations in terms of unit vectors

Let $$\overrightarrow a$$ = a1$$\overrightarrow i$$ + a2$$\overrightarrow j$$ and $$\overrightarrow b$$ = b1$$\overrightarrow i$$ + b2$$\overrightarrow j$$

1. Addition of vectors:
$$\overrightarrow a$$ + $$\overrightarrow b$$ = a1$$\overrightarrow i$$ + a2$$\overrightarrow j$$ + b1$$\overrightarrow i$$ + b2$$\overrightarrow j$$ = (a1 + b1)$$\overrightarrow i$$ + (a2 + b2)$$\overrightarrow j$$
2. Substraction of Vectors:
$$\overrightarrow a$$ - $$\overrightarrow b$$ = a1$$\overrightarrow i$$ + a2$$\overrightarrow j$$ - (b1$$\overrightarrow i$$ + b2$$\overrightarrow j$$) =a1$$\overrightarrow i$$ + a2$$\overrightarrow j$$ - b1$$\overrightarrow i$$ - b2$$\overrightarrow j$$
3. Scalar product of vectors:
$$\overrightarrow a . \overrightarrow b = (a_1\overrightarrow i + a_2\overrightarrow j) . (b_1\overrightarrow i + b_2\overrightarrow j) = a_1b_1 \overrightarrow i . \overrightarrow i + a_1b_2 \overrightarrow i . \overrightarrow j + a_2b_1 \overrightarrow j . \overrightarrow i + a_2b_2 \overrightarrow j . \overrightarrow j = a_1b_1 + 0 + 0 + a+2b_2 = a_1b_1 + a_2b_2$$

#### Magnitude and direction of a vector in terms of unit vectors

Let $$\overrightarrow a$$ = x$$\overrightarrow i$$ + y$$\overrightarrow j$$ be a vector.

Then,

$$\overrightarrow a$$ = $$\begin{pmatrix}x\\ y\\ \end{pmatrix}$$

∴ X- component of $$\overrightarrow a$$ = x and Y-component of $$\overrightarrow a$$ = y

Magnitude of $$\overrightarrow a$$

|$$\overrightarrow a$$| = $$\sqrt {x^2 + y^2}$$ = $$\sqrt {(coefficient\;of\;\overrightarrow i)^2 + (coefficient\;of\;\overrightarrow j)^2}$$

Direction of $$\overrightarrow a$$

tan$$\theta$$ = $$\frac yx$$ = $$\frac {coefficient\;of\;\overrightarrow j}{coefficient\;of\;\overrightarrow i}$$

Unit vector along the direction of $$\overrightarrow a$$

$$\widehat a$$ = $$\frac {\overrightarrow a}{|\overrightarrow a|}$$ = $$\frac {1}{\sqrt {x^2 + y^2}}$$ (x$$\overrightarrow i$$ + y$$\overrightarrow j$$) = $$\frac {x\overrightarrow i + y\overrightarrow j}{\sqrt {x^2 + y^2}}$$

#### Vector Geometry

Theorem 1: (Mid- point Formula)

If $$\overrightarrow a$$ and $$\overrightarrow b$$ are position vector of two points A and B respectively and M is the middle point of the line segment AB, then the position vector of M is $$\frac 12$$($$\overrightarrow a$$ + $$\overrightarrow b$$).

Proof:

Let AB be a line segment and O be the origin.

Here,

Position Vector of A = $$\overrightarrow {OA}$$ = $$\overrightarrow a$$

Position Vector of B = $$\overrightarrow {OB}$$ = $$\overrightarrow b$$

Let M be the middle point of the segment AB.

Then,

\begin{align*} \overrightarrow {OM} &= \overrightarrow {OA} + \overrightarrow {AM}\\ &= \overrightarrow {OA} + \frac 12\overrightarrow {AB}\\ &= \overrightarrow {OA} + \frac 12(\overrightarrow {OB} - \overrightarrow {OA})\\ &= \overrightarrow a + \frac 12(\overrightarrow b - \overrightarrow a)\\ &= \frac {2\overrightarrow a + \overrightarrow b - \overrightarrow a}{2}\\ &= \frac 12(\overrightarrow a + \overrightarrow b)\\ \end{align*}

∴ Position Vector of M = $$\frac 12$$($$\overrightarrow a$$ + $$\overrightarrow b$$) Proved

Theorem 2: (Section Formula for Internal Division)

If $$\overrightarrow a$$ and $$\overrightarrow b$$ are position vector of two points A and B respectively and the point M divides the line segment AB internally in the ratio m : n, then the position vector of M is $$\frac {m \overrightarrow b + n\overrightarrow a}{m + n}$$.

Proof:

Let AB be a line segment and O be the origin.

Here,

Position Vector of A = $$\overrightarrow {OA}$$ = $$\overrightarrow a$$

Position Vector of B = $$\overrightarrow {OB}$$ = $$\overrightarrow b$$

Let the point M divides AB internally in the ratio m : n.

Then,

\begin{align*} \overrightarrow {OM} &= \overrightarrow {OA} + \overrightarrow {AM}\\ &= \overrightarrow {OA} + \frac {m}{m + n}\overrightarrow {AB}\\ &= \overrightarrow {OA} + \frac m{m + n} (\overrightarrow {OB} - \overrightarrow {OA})\\ &= \overrightarrow a + \frac m{m + n} (-\overrightarrow a)\\ &= \frac {m\overrightarrow a + n\overrightarrow a + m\overrightarrow b - m\overrightarrow a}{m + n}\\ &= \frac {m\overrightarrow b + n\overrightarrow a}{m + n}\\ \end{align*}

∴ Position Vector of M = $$\frac {m\overrightarrow b + n\overrightarrow a}{m + n}$$ Proved

Theorem 3: (Section Formula for External Division)

If $$\overrightarrow a$$ and $$\overrightarrow b$$ are position vector of two points A and B respectively and the point P divides the line segment AB externally in the ratio m : n, then the position vector of P is $$\frac {m \overrightarrow b - n\overrightarrow a}{m - n}$$.

Proof:

Let AB be a line segment and O be the origin.

Here,

Position Vector of A = $$\overrightarrow {OA}$$ = $$\overrightarrow a$$

Position Vector of B = $$\overrightarrow {OB}$$ = $$\overrightarrow b$$

Let the point P divides AB externally in the ratio m : n.

Then,

$$\frac {AP}{BP}$$ = $$\frac mn$$

∴ n$$\overrightarrow {AP}$$ = m$$\overrightarrow {BP}$$

or, n ($$\overrightarrow {OP}$$ - $$\overrightarrow {OA}$$) = m($$\overrightarrow {OP}$$ - $$\overrightarrow {OB}$$)

or, n$$\overrightarrow {OP}$$ - n$$\overrightarrow {OA}$$ = m$$\overrightarrow {OP}$$ - m$$\overrightarrow {OB}$$

or, m$$\overrightarrow {OB}$$ - n$$\overrightarrow {OA}$$ = m$$\overrightarrow {OP}$$ - n$$\overrightarrow {OP}$$

or, m$$\overrightarrow b$$ - n$$\overrightarrow a$$ = (m - n)$$\overrightarrow {OP}$$

∴ $$\overrightarrow {OP}$$ = $$\frac {m\overrightarrow b - n\overrightarrow a}{m - n}$$

So, the position vector of P = $$\frac {m\overrightarrow b - n\overrightarrow a}{m - n}$$ Proved

Theorem 4:

The line segment joining the mid-point of two sides of a triangle is parallel to the third side and it is half of it.

Proof:

Let ABC be a triangle and P and Q be the mid-points of the sides AB and AC respectively.

Here,

$$\overrightarrow {PA}$$ = $$\frac 12$$$$\overrightarrow {BA}$$

$$\overrightarrow {AQ}$$ = $$\frac 12$$$$\overrightarrow {AC}$$

Then,

$$\overrightarrow {PA}$$ + $$\overrightarrow {AQ}$$ = $$\frac 12$$$$\overrightarrow {BA}$$ + $$\frac 12$$$$\overrightarrow {AC}$$

∴ $$\overrightarrow {PQ}$$ = $$\frac 12$$($$\overrightarrow {BA}$$ + $$\overrightarrow {AC}$$)

i.e. $$\overrightarrow {PQ}$$ = $$\frac 12$$$$\overrightarrow {BC}$$

Clearly, $$\overrightarrow {PQ}$$ // $$\overrightarrow {BC}$$ Proved

Theorem 5:

The position vector of the centroid of a triangle is given by $$\overrightarrow g$$ = $$\frac 13$$ ($$\overrightarrow a$$ + $$\overrightarrow b$$ + $$\overrightarrow c$$) where $$\overrightarrow a$$, $$\overrightarrow b$$ and $$\overrightarrow c$$ are the position vectors of the vertices and $$\overrightarrow g$$ is the position vector of the centroid.

Proof:

Let ABC be triangle and O be the origin.

Let,

$$\overrightarrow {OA}$$ = $$\overrightarrow a$$

$$\overrightarrow {OB}$$ = $$\overrightarrow b$$

$$\overrightarrow {OC}$$ = $$\overrightarrow c$$

Let D be the mid- point of AC.

By mid-point theorem,

$$\overrightarrow {OD}$$ = $$\frac 12$$($$\overrightarrow a$$ + $$\overrightarrow c$$)

Let G be the centroid of the triangle ABC.

Then G divides BD internally in the ratio 2: 1.

Then,

\begin{align*} \overrightarrow {OG} &= \frac {2\overrightarrow {OD} + 1\overrightarrow {OB}}{2 + 1}\\ &= \frac {2 × \frac 12 \overrightarrow a + \overrightarrow c + \overrightarrow b}{3}\\ &= \frac 13 (\overrightarrow a + \overrightarrow b + \overrightarrow c)\\ \end{align*}

∴ Position Vector of G ($$\overrightarrow g$$) = $$\frac 13 (\overrightarrow a + \overrightarrow b + \overrightarrow c)$$ Proved

Theorem 6:

The median to the base of an isosceles triangle is perpendicular to the base.

ABC be an isosceles triangle where AB = ACand AM is the median to the base BC.Proof:

Let $$\overrightarrow {AB}$$ = $$\overrightarrow a$$ and $$\overrightarrow {AC}$$ = $$\overrightarrow b$$

Then,

|$$\overrightarrow a$$| = |$$\overrightarrow b$$| or a = b

Now,

$$\overrightarrow {AM}$$ = $$\overrightarrow {AB}$$ + $$\overrightarrow {BM}$$ = $$\overrightarrow {AB}$$ + $$\frac 12$$$$\overrightarrow {BC}$$

$$\overrightarrow {BC}$$ = $$\overrightarrow {BA}$$ + $$\overrightarrow {AC}$$ = -$$\overrightarrow a$$ + $$\overrightarrow b$$

\begin{align*} \therefore \overrightarrow {AM} &= \overrightarrow a + \frac 12 (\overrightarrow b - \overrightarrow a)\\ &= \frac {\overrightarrow a + \overrightarrow b - \overrightarrow c}2\\ &= \frac 12(\overrightarrow a + \overrightarrow b)\\ \end{align*}

Now,

\begin{align*} \overrightarrow {AM} . \overrightarrow {BC} &= \frac 12(\overrightarrow a + \overrightarrow b) . (\overrightarrow b - \overrightarrow a)\\ &= \frac 12 (\overrightarrow b + \overrightarrow a) . (\overrightarrow b - \overrightarrow a)\\ &= \frac 12(b^2 - a^2)\\ &= 0\\ \end{align*}

∴ AM⊥ BC Proved

Theorem 7:

The figure formed by joining the mid-point of the adjacent sides of a quadrilateral is a parallelogram.

Proof:

Let ABCD be a quadrilateral and P, Q, R and S are the mid-points of AB, BC, CD and DA respectively. Join BD.

Then,

$$\overrightarrow {BA}$$ + $$\overrightarrow {AD}$$ = $$\overrightarrow {BD}$$

2$$\overrightarrow {PA}$$ + 2$$\overrightarrow {AS}$$ = $$\overrightarrow {BD}$$

$$\overrightarrow {PA}$$ + $$\overrightarrow {AS}$$ = $$\frac 12$$$$\overrightarrow {BD}$$

∴ $$\overrightarrow {PS}$$ = $$\frac 12$$$$\overrightarrow {BD}$$

Clearly, $$\overrightarrow {PS}$$ // $$\overrightarrow {BD}$$

Again,

$$\overrightarrow {BC}$$ + $$\overrightarrow {CD}$$ = $$\overrightarrow {BD}$$

2$$\overrightarrow {QC}$$ + 2$$\overrightarrow {CR}$$ = $$\overrightarrow {BD}$$

$$\overrightarrow {QC}$$ + $$\overrightarrow {CR}$$ = $$\frac 12$$$$\overrightarrow {BD}$$

∴ $$\overrightarrow {QC}$$ = $$\frac 12$$$$\overrightarrow {BD}$$

Clearly, $$\overrightarrow {QR}$$ // $$\overrightarrow {BD}$$

$$\overrightarrow {PS}$$ = $$\overrightarrow {QR}$$, $$\overrightarrow {PS}$$ // $$\overrightarrow {QR}$$

Similarly,

$$\overrightarrow {PQ}$$ = $$\overrightarrow {SR}$$, $$\overrightarrow {PQ}$$ // $$\overrightarrow {SR}$$

Hence, PQRS is a parallelogram.

Theorem 8:

The diagonals of a parallelogram bisect each other.

Proof:

Let OACB be a parallelogram and O be the origin.

Let $$\overrightarrow {OA}$$ = $$\overrightarrow a$$ and $$\overrightarrow {OB}$$ = $$\overrightarrow b$$

Let M be the middle point of AB.

Then,

$$\overrightarrow {OM}$$ = $$\frac 12$$($$\overrightarrow a$$ + $$\overrightarrow b$$).............................................(i)

Also,

$$\overrightarrow {OC}$$ = $$\overrightarrow a$$ + $$\overrightarrow b$$, by parallelogram of vector addition.

Let N be the middle point of OC.

Then,

$$\overrightarrow {ON}$$ = $$\frac 12$$$$\overrightarrow {OC}$$ = $$\frac 12$$($$\overrightarrow a$$ + $$\overrightarrow b$$)............................................(ii)

From (i) and (ii), M and N have the same position vector. So, M and N are same points.

Hence, the diagonals of a parallelogram bisect each other.

Theorem 9:

The diagonals of a rectangle are equal.

Proof:

Let ABCD be a rectangle and AC and BD are diagonals.

Then,

$$\overrightarrow {AB}$$ = $$\overrightarrow {DC}$$, $$\overrightarrow {AD}$$ = $$\overrightarrow {BC}$$, $$\angle A$$ = $$\angle B$$ = $$\angle C$$ = $$\angle D$$ = 90° ($$\overrightarrow {AB}$$ . $$\overrightarrow {BC}$$ = 0, $$\overrightarrow {BA}$$ . $$\overrightarrow {AD}$$ = 0)

Now,

$$\overrightarrow {AC}$$ = $$\overrightarrow {AB}$$ + $$\overrightarrow {BC}$$

$$\overrightarrow {AC}^2$$ = ($$\overrightarrow {AB}$$ + $$\overrightarrow {BC}$$)2

AC2 = $$\overrightarrow {AB}^2$$ + 2$$\overrightarrow {AB}$$ . $$\overrightarrow {BC}$$ + $$\overrightarrow {BC}^2$$

AC2= AB2 + 0 + BC2

AC2= AB2 + BC2......................................(i)

Again,

$$\overrightarrow {BD}$$ = $$\overrightarrow {BA}$$ + $$\overrightarrow {AD}$$

$$\overrightarrow {BD}^2$$ = ($$\overrightarrow {BA}$$ + $$\overrightarrow {AD}$$)2

BD2= $$\overrightarrow {BA}^2$$ + 2$$\overrightarrow {BA}$$ . $$\overrightarrow {AD}$$ + $$\overrightarrow {AD}^2$$

BD2= BA2 + 0 + AD2

BD2= AB2 + BC2......................................(i)

From (i) and (ii),

AC2= BD2

∴ AC = BD

Hence, the diagonals of a rectangle are equal.

Theorem 10:

The diagonals of a rhombus bisect each other at right angles.

Proof:

Let ABCD is a rhombus and AC and BD are diagonals.

Obviously,

$$\overrightarrow {AO}$$ = $$\overrightarrow {OC}$$ and $$\overrightarrow {BO}$$ = $$\overrightarrow {OD}$$

Now,

We prove that,

$$\overrightarrow {AO}$$⊥ $$\overrightarrow {BO}$$

Let $$\overrightarrow {AB}$$ = $$\overrightarrow a$$ and $$\overrightarrow {AD}$$ = $$\overrightarrow b$$

Then,

$$\overrightarrow {AC}$$ = $$\overrightarrow {AB}$$ + $$\overrightarrow {BC}$$ = $$\overrightarrow {AB}$$ + $$\overrightarrow {AD}$$ = $$\overrightarrow a$$ + $$\overrightarrow b$$

$$\overrightarrow {AO}$$ = $$\frac 12$$$$\overrightarrow {AC}$$ = $$\frac 12$$($$\overrightarrow a$$ + $$\overrightarrow b$$)..............................(i)

Again,

$$\overrightarrow {BD}$$ = $$\overrightarrow {BA}$$ + $$\overrightarrow {AD}$$ = -$$\overrightarrow a$$ + $$\overrightarrow b$$ = $$\overrightarrow b$$ - $$\overrightarrow a$$

$$\overrightarrow {BO}$$ = $$\frac 12$$$$\overrightarrow {BD}$$ = $$\frac 12$$($$\overrightarrow b$$ - $$\overrightarrow a$$)..............................(ii)

From (ii) and (ii),

$$\overrightarrow {AO}$$ .$$\overrightarrow {BO}$$ =$$\frac 12$$($$\overrightarrow a$$ + $$\overrightarrow b$$ .$$\frac 12$$($$\overrightarrow b$$ - $$\overrightarrow a$$) = $$\frac 14$$ ($$\overrightarrow b^2$$ - $$\overrightarrow a^2$$) = $$\frac 14$$(b2 - a2) = 0

Hence, the diagonals of a rhombus bisect each other at right angles.

Theorem 11:

The angle in a semi- circle is a right angle.

Proof:

Let O be the centre of a circle and AB be a diameter. $$\angle$$ACB is an angle in the semi- circle. Join OC.

Now,

$$\overrightarrow {AC}$$ = $$\overrightarrow {AO}$$ + $$\overrightarrow {OC}$$..............................(i)

$$\overrightarrow {CB}$$ = $$\overrightarrow {CO}$$ + $$\overrightarrow {OB}$$ = $$\overrightarrow {CO}$$ + $$\overrightarrow {AO}$$

$$\overrightarrow {CB}$$ = $$\overrightarrow {AO} - \overrightarrow {OC}$$...................................(ii)

From (i) and (ii),

\begin{align*} \overrightarrow {AC} . \overrightarrow {CB} &= (\overrightarrow {AO} + \overrightarrow {OC}) . (\overrightarrow {AO} - \overrightarrow {OC})\\ & =(\overrightarrow {AO}^2 - \overrightarrow {OC}^2\\ &= AO^2 - OC^2 [since\; AO = OC]\\ &= 0\\ \end{align*}

Hence, the angle in the semi- circle is a right angle.

Theorem 12:

The mid- point of the hypotenuse of a right-angled triangle is equidistant from its vertices.

Proof:

Let AOC be a right angled triangle and O be the origin.

Let $$\angle AOC$$ = 90° and M is the mid- point of AC.

Then,

$$\overrightarrow {OM}$$ = $$\frac {\overrightarrow {OA} + \overrightarrow {OC}}2$$

2$$\overrightarrow {OM}$$ = $$\overrightarrow {OA}$$ + $$\overrightarrow {OC}$$

(2$$\overrightarrow {OM}$$)2 = ($$\overrightarrow {OA}$$ + $$\overrightarrow {OC}$$)2

4 $$\overrightarrow {OM}^2$$ = $$\overrightarrow {OA}^2$$ + 2$$\overrightarrow {OA}$$ . $$\overrightarrow {OC}$$ + $$\overrightarrow {OC}^2$$

4 OM2 = OA2 + 0 + OC2

4 OM2 = OA2 + OC2

4 OM2 = AC2

OM2 = $$\frac {AC^2}4$$

∴ OM = $$\frac 12$$AC

Also,

AM = MC = $$\frac 12$$AC

∴ AM = MC = OM

Hence, the mid- point of the hypotenuse of a right angled triangle is equidistant from its vertices.

Dot product of $$\overrightarrow a$$ & $$\overrightarrow b$$  = $$\overrightarrow a$$ . $$\overrightarrow b$$
If $$\overrightarrow i$$ and $$\overrightarrow j$$ are unit vectors along x-axis and y-axis $$\overrightarrow i$$.$$\overrightarrow i$$ = $$\overrightarrow j$$. $$\overrightarrow j$$ = 1 and $$\overrightarrow i$$.$$\overrightarrow j$$ = $$\overrightarrow j$$.$$\overrightarrow i$$ = 0

.

### Very Short Questions

Here,

$$\vec a$$ = $$\begin {pmatrix} 6\\ 1\\ \end {pmatrix}$$ and $$\vec b$$ = $$\begin {pmatrix} -1\\ 6\\ \end {pmatrix}$$

$$\vec a$$ . $$\vec b$$ = x1x2 + y1y2 = 6× -1 + 1× 6 = 6 - 6 = 0

$$\begin {vmatrix} \vec a\\ \end {vmatrix}$$ = $$\sqrt {x_1^2 + y_1^2}$$ = $$\sqrt {(6)^2 + (1)^2}$$ = $$\sqrt {36 + 1}$$ = $$\sqrt {37}$$

$$\begin {vmatrix} \vec b\\ \end {vmatrix}$$ = $$\sqrt {x_1^2 + y_1^2}$$ = $$\sqrt {(-1)^2 + (6)^2}$$ = $$\sqrt {1 + 36}$$ = $$\sqrt {37}$$

cos$$\theta$$ = $$\frac {\vec a . \vec b}{\begin {vmatrix} \vec a\\ \end {vmatrix} \begin {vmatrix} \vec b\\ \end {vmatrix}}$$

or, cos$$\theta$$ = $$\frac 0{\sqrt {37} \sqrt {37}}$$

or, cos$$\theta$$ = 0

or, $$\theta$$ = cos-1(0) = 90°

The angle between two vectors is 90° so, the $$\vec a$$ and $$\vec b$$ are perpendicular each other. Proved

Here,

$$\vec a$$ = $$\begin {pmatrix} 2\\ 1\\ \end {pmatrix}$$ and $$\vec b$$ = $$\begin {pmatrix} 0\\ -2\\ \end {pmatrix}$$

$$\vec a$$ . $$\vec b$$ = x1x2 + y1y2 = 2 × 0 + 1 × -2 = 0 + -2 = -2

$$\begin {vmatrix} \vec a\\ \end {vmatrix}$$ = $$\sqrt {x^2 + y^2}$$ = $$\sqrt {(2)^2 + (1)^2}$$ = $$\sqrt {4 + 1}$$ = $$\sqrt 5$$

$$\begin {vmatrix} \vec b\\ \end {vmatrix}$$ = $$\sqrt {x^2 + y^2}$$ = $$\sqrt {(0)^2 + (-2)^2}$$ = $$\sqrt {0 + 4}$$ = 2

cos$$\theta$$ = $$\frac {-2}{2\sqrt 5}$$ = -$$\frac 1{\sqrt 5}$$

∴ $$\theta$$ = cos-1 (-$$\frac 1{\sqrt 5}$$) Ans

Here,

$$\vec p$$ = $$\begin {pmatrix} 3\\ 4\\ \end {pmatrix}$$ and $$\vec q$$ = $$\begin {pmatrix} 1\\ 1\\ \end {pmatrix}$$

$$\vec p$$ . $$\vec q$$ = x1x2 + y1y2 = 3 × 1 + 4 × 1 = 3 + 4 = 7

$$\begin {vmatrix} \vec p\\ \end {vmatrix}$$ = $$\sqrt {x^2 + y^2}$$ = $$\sqrt {(3)^2 + (4)^2}$$ = $$\sqrt {9 + 16}$$ = $$\sqrt 25$$ = 5

$$\begin {vmatrix} \vec q\\ \end {vmatrix}$$ = $$\sqrt {x^2 + y^2}$$ = $$\sqrt {(1)^2 + (1)^2}$$ = $$\sqrt {1 + 1}$$ = $$\sqrt 2$$

cos$$\theta$$ = $$\frac {7}{5\sqrt 2}$$ = $$\frac 7{7.07}$$ = 0.99

∴ $$\theta$$ = cos-1 (0.99) = 8.110Ans

Here,

$$\vec a$$ = $$\begin {pmatrix} 2\\ -3\\ \end {pmatrix}$$ and $$\vec c$$ = $$\begin {pmatrix} 4\\ -2\\ \end {pmatrix}$$

$$\vec a$$ . $$\vec c$$ = x1x2 + y1y2 = 2× 4+ (-3) × (-2) = 8+ 6=14

$$\begin {vmatrix} \vec a\\ \end {vmatrix}$$ = $$\sqrt {x^2 + y^2}$$ = $$\sqrt {(2)^2 + (-3)^2}$$ = $$\sqrt {4 + 9}$$ = $$\sqrt {13}$$

$$\begin {vmatrix} \vec c\\ \end {vmatrix}$$ = $$\sqrt {x^2 + y^2}$$ = $$\sqrt {(4)^2 + (-2)^2}$$ = $$\sqrt {16 + 4}$$ = $$\sqrt {20}$$

cos$$\theta$$ = $$\frac {14}{\sqrt {13} . \sqrt {20}}$$ = $$\frac {14}{16.13}$$ = 0.87

∴ $$\theta$$ = cos-1 (0.87) = 29.540Ans

Here,

$$\begin {vmatrix} \vec p\\ \end {vmatrix}$$ = 3, $$\begin {vmatrix} \vec q\\ \end {vmatrix}$$ = 3$$\sqrt 2$$ and $$\vec p$$.$$\vec q$$ = 9

We know,

cos$$\theta$$ = $$\frac {\vec p . \vec q}{\begin {vmatrix} \vec p\\ \end {vmatrix}\begin {vmatrix} \vec q\\ \end {vmatrix}}$$ = $$\frac 9{3 × 3\sqrt 2}$$ = $$\frac 1{\sqrt 2}$$

cos$$\theta$$ = $$\frac 1{\sqrt 2}$$

$$\theta$$ = cos-1($$\frac 1{\sqrt 2}$$) = 45°

∴ The angle between $$\vec p$$ and $$\vec q$$ is 45°. Ans

Here,

$$\vec a$$ = 4 $$\vec i$$ + $$\vec j$$ and $$\vec b$$ = 3 $$\vec i$$ + 4$$\vec j$$

(i)

$$\vec a$$.$$\vec b$$ = (4 $$\vec i$$ + $$\vec j$$) . (3$$\vec i$$ + 4$$\vec j$$) = 4× 3 + 1× 4 = 12 + 4 = 16 Ans

(ii)

$$\begin {vmatrix} \vec a\\ \end {vmatrix}$$ = $$\sqrt {x_1^2 + y_1^2}$$ = $$\sqrt {(4)^2 + (1)^2}$$ = $$\sqrt {16 + 1}$$ = $$\sqrt {17}$$

$$\begin {vmatrix} \vec b\\ \end {vmatrix}$$ = $$\sqrt {x_1^2 + y_1^2}$$ = $$\sqrt {(3)^2 + (4)^2}$$ = $$\sqrt {9 + 16}$$ = $$\sqrt {25}$$ = 5

Let: $$\theta$$ be the angle between $$\vec a$$ and $$\vec b$$ then,

cos$$\theta$$ = $$\frac {\vec a . \vec b}{\begin {vmatrix} \vec a\\ \end {vmatrix} \begin {vmatrix} \vec b\\ \end {vmatrix}}$$ = $$\frac {16}{5\sqrt {17}}$$

$$\theta$$ = cos-1($$\frac {16}{5\sqrt {17}}$$) = 37.47°

∴ The angle between $$\vec a$$ and $$\vec b$$ is 37.47°. Ans

Here,

$$\vec a$$ = -4$$\vec i$$ + 5$$\vec j$$ and $$\vec b$$ = 8$$\vec i$$ - 10$$\vec j$$

(x1 , y1) = (-4 , 5) and (x2 , y2) = (8 , -10)

$$\vec a$$.$$\vec b$$ = x1x2 + y1y2 = -4 × 8 + 5 × 10 = -32 - 50 = -82

$$\begin {vmatrix} \vec a\\ \end {vmatrix}$$ = $$\sqrt {x_1^2 + y_1^2}$$ = $$\sqrt {(-4)^2 + (5)^2}$$ = $$\sqrt {16 + 25}$$ = $$\sqrt {41}$$

$$\begin {vmatrix} \vec b\\ \end {vmatrix}$$ = $$\sqrt {x_1^2 + y_1^2}$$ = $$\sqrt {(8)^2 + (-10)^2}$$ = $$\sqrt {64 + 100}$$ = $$\sqrt {164}$$

We know,

cos$$\theta$$ = $$\frac {\vec a . \vec b}{\begin {vmatrix} \vec a\\ \end {vmatrix} \begin {vmatrix} \vec b\\ \end {vmatrix}}$$ = $$\frac {-82}{\sqrt {41} \sqrt {164}}$$ = - 1 (Approx.)

$$\theta$$ = cos-1 (-1) = 180°

∴ The $$\vec a$$ and $$\vec b$$ are parallel in opposite direction. Proved

Here,

$$\vec a$$ = 3 + k and $$\vec b$$ = -7 + 3 are perpendiculat each other.

Then:

$$\vec a$$ . $$\vec b$$ = 0

or, (3 + k) (-7 + 3) = 0

or, 3× -7 + k× 3 = 0

or, -21 + 3k = 0

or, 3k = 21

or, k = $$\frac {21}3$$

∴ k = 7 Ans

Here,

$$\vec a$$ = 4 + 2 and $$\vec b$$ = -1 + 2

If $$\vec a$$ and $$\vec b$$ are perpendicular than $$\vec a$$ . $$\vec b$$ = 0

or, (4 + 2) (-1 + 2) = 0

or, 4× -1 + 2× 2 = 0

or, -4 + 4 = 0

∴ 0 = 0

∴ $$\vec a$$ and $$\vec b$$ are perpendicular to each other. Proved

Here,

$$\vec p$$ = 10$$\vec i$$ + 2k $$\vec j$$ and $$\vec q$$ = 2$$\vec i$$ - 5$$\vec j$$

(x1, y1) = (10, 2k) and (x2, y2) = (2. -5)

We know,

$$\vec p$$ and $$\vec q$$ are perpendicular.

Then: $$\vec p$$ . $$\vec q$$ = 0

or, (10, 2k) (2, -5) = 0

or, 10× 2 + 2k× -5 = 0

or, 20 - 10k = 0

or, 10k = 20

or, k = $$\frac {20}{10}$$

∴ k = 2 Ans

Here,

$$\vec {OA}$$ = 7$$\vec i$$ - 5$$\vec j$$ and $$\vec {OB}$$ = 5$$\vec i$$ - 7$$\vec j$$

$$\vec a$$ . $$\vec b$$ = x1x2 + y1y2 = 7 × 5 + (-5) × (-7) = 35 + 35 = 70

$$\begin {vmatrix} \vec a\\ \end {vmatrix}$$ = $$\sqrt {x_1^2 + y_1^2}$$ = $$\sqrt {(7)^2 + (-5)^2}$$ = $$\sqrt {49 + 25}$$ = $$\sqrt {74}$$

$$\begin {vmatrix} \vec b\\ \end {vmatrix}$$ = $$\sqrt {x_1^2 + y_1^2}$$ = $$\sqrt {(5)^2 + (-7)^2}$$ = $$\sqrt {25 + 49}$$ = $$\sqrt {74}$$

cos$$\theta$$ = $$\frac {\vec a . \vec b}{\begin {vmatrix} \vec a\\ \end {vmatrix} \begin {vmatrix} \vec b\\ \end {vmatrix}}$$ =$$\frac {70}{\sqrt {74} \sqrt {74}}$$ = $$\frac {70}{74}$$ = 0.95

$$\theta$$ = cos-1(0.95) = 18.93°

∴$$\angle$$AOB = 18.93° Ans

Let: the vector $$\vec a$$ and $$\vec b$$ are parallel if $$\vec a$$ = m$$\vec b$$ where m is scalar quantity.

$$\vec a$$ = 3$$\vec i$$ + 4$$\vec j$$ and b = 8$$\vec i$$ - 6$$\vec j$$

$$\vec a$$ . $$\vec b$$ = x1x2 + y1y2 = 3 × 8 + 4 × (-6) = 24 - 24 = 0

$$\begin {vmatrix} \vec a\\ \end {vmatrix}$$ = $$\sqrt {x_1^2 + y_1^2}$$ = $$\sqrt {(3)^2 + (4)^2}$$ = $$\sqrt {9 + 16}$$ = $$\sqrt {25}$$ = 5

$$\begin {vmatrix} \vec b\\ \end {vmatrix}$$ = $$\sqrt {x_1^2 + y_1^2}$$ = $$\sqrt {(8)^2 + (-6)^2}$$ = $$\sqrt {64 + 36}$$ = $$\sqrt {100}$$ = 10

cos$$\theta$$ = $$\frac {\vec a . \vec b}{\begin {vmatrix} \vec a\\ \end {vmatrix} \begin {vmatrix} \vec b\\ \end {vmatrix}}$$ = $$\frac 0{5 × 10}$$ = 0

∴ $$\theta$$ = cos-1 (0) = 90° Ans

Here,

$$\vec {OA}$$ = $$\vec a$$, $$\vec {OB}$$ = $$\vec b$$ and $$\vec {AC}$$ = 3$$\vec {AB}$$ then, $$\vec {OC}$$ = ?

Using triangle law in vector addition,

$$\vec {OA}$$ + $$\vec {AB}$$ = $$\vec {OB}$$

$$\vec {AB}$$ = $$\vec {OB}$$ - $$\vec {OA}$$ = $$\vec b$$ - $$\vec a$$

We know,

$$\vec {AC}$$ = 2$$\vec {AB}$$ = 3 ($$\vec b$$ - $$\vec a$$) = 3$$\vec b$$ - 3$$\vec a$$

Again,

$$\vec {OA}$$ + $$\vec {AC}$$ = $$\vec {OC}$$

∴ $$\vec {OC}$$ = $$\vec a$$ + 3$$\vec b$$ - 3$$\vec a$$ = 3$$\vec b$$ - 2$$\vec a$$ Ans

Given points are: A(1, 2) and B(3, 0)

x-component = x2 - x1 = 3 - 1 = 2

y-component = y2 - y1 = 0 - 2 = - 2

$$\vec {AB}$$ = (2, -2)

$$\vec {AB}$$ = 2$$\vec i$$ - 2$$\vec j$$ Ans

Given points are: C(1, 1) and D(-2, -4)

x-component = x2 - x1 = -2 - 1 = -3

y-component = y2 - y1 = 0 - 2 = - 2

$$\vec {CD}$$ = (-3, -5)

$$\vec {CD}$$ = -3$$\vec i$$ - 5$$\vec j$$ Ans

Given points are: P(3, 5) and Q(-7, 3).

Let, O be the origin.

$$\vec {OP}$$ = $$\begin {pmatrix} 3\\ 5\\ \end {pmatrix}$$ and $$\vec {OQ}$$ = $$\begin {pmatrix} -7\\ 3\\ \end {pmatrix}$$

Using midpoint formula,

$$\vec {OM}$$ = $$\frac {\vec {OP} + \vec {OQ}}2$$

or, $$\vec {OM}$$ = $$\frac {\begin {pmatrix}3\\ 5\\ \end{pmatrix} + \begin {pmatrix} -7\\ 3\\ \end {pmatrix}}2$$

or, $$\vec {OM}$$ = $$\frac {\begin {pmatrix}3 - 7\\ 5 + 3\\ \end {pmatrix}}2$$

or, $$\vec {OM}$$ = $$\frac {\begin {pmatrix} -4\\ 3\\ \end {pmatrix}}2$$

or, $$\vec {OM}$$ = $$\begin {pmatrix} \frac {-4}2\\ \frac 82\\ \end {pmatrix}$$

∴ $$\vec {OM}$$ = $$\begin {pmatrix} -2\\ 4\\ \end {pmatrix}$$

∴ $$\vec {OM}$$ = -2$$\vec i$$ + 4$$\vec j$$ Ans

Here,

D is the midpoint of the line BC of the $$\triangle$$ABC.

In $$\triangle$$ABD,

$$\vec {AB}$$ = $$\vec {AD}$$ + $$\vec {DB}$$....................(1) [$$\because$$ triangle law of vector addition]

In $$\triangle$$ACD,

$$\vec {AC}$$ = $$\vec {AD}$$ + $$\vec {DC}$$...................(2)[$$\because$$ triangle law of vector addition]

Adding (1) and (2)

$$\vec {AB}$$ + $$\vec {AC}$$ = $$\vec {AD}$$ + $$\vec {BD}$$ + $$\vec {AD}$$ + $$\vec {DC}$$

or,$$\vec {AB}$$ + $$\vec {AC}$$ = 2$$\vec {AD}$$ + ($$\vec {DB}$$ + $$\vec {DC}$$)

or,$$\vec {AB}$$ + $$\vec {AC}$$ = 2$$\vec {AD}$$ + $$\vec {DB}$$ - $$\vec {DB}$$ [$$\because$$ -$$\vec {DB}$$ = $$\vec {DC}$$]

∴ $$\vec {AB}$$ + $$\vec {AC}$$ = 2$$\vec {AD}$$ Ans

Let, O be the origin.

$$\vec {OA}$$ = 3$$\vec i$$ + 2$$\vec j$$ and $$\vec {OB}$$ = 5$$\vec i$$ - 6$$\vec j$$

We know,

$$\vec {OM}$$ = $$\frac {\vec {OA} + \vec {OB}}2$$

or, $$\vec {OM}$$ = $$\frac {3\vec i + 2\vec j + 5\vec i - 6\vec j}2$$

or, $$\vec {OM}$$ = $$\frac {8\vec i - 4\vec j}2$$

∴ $$\vec {OM}$$ = 4$$\vec i$$ - 2$$\vec j$$

∴ The position vector of M = 4$$\vec i$$ - 2$$\vec j$$ Ans

Let: O be the origin.

$$\vec {OA}$$ = 5$$\vec i$$ + 2$$\vec j$$ and $$\vec {OB}$$ = 3$$\vec i$$ + 6$$\vec j$$

m : n = 2 : 3

Using section internal division formula.

$$\vec {OP}$$ = $$\frac {m \vec{OB} + n\vec {OA}}{m + n}$$

or, $$\vec {OP}$$ = $$\frac {2(3\vec i + 6\vec j) + 3(5\vec i + 2\vec j)}{2 + 3}$$

or, $$\vec {OP}$$ = $$\frac {6\vec i + 12\vec j + 15\vec i + 6\vec j}5$$

or, $$\vec {OP}$$ = $$\frac {21\vec i + 18\vec j}5$$

∴ $$\vec {OP}$$ = $$\frac {21}5$$$$\vec i$$ + $$\frac {18}5$$$$\vec j$$ Ans

Here,

A(-4, 8) and B(3, 7)

Let, O be the origin.

$$\vec {OA}$$ = $$\begin {pmatrix} -4\\ 8\\ \end {pmatrix}$$

$$\vec {OB}$$ = $$\begin {pmatrix} 3\\ 7\\ \end {pmatrix}$$

Using section external division formula,

$$\vec {AB}$$ = $$\frac {4\begin {pmatrix} 3\\ 7\\ \end {pmatrix} - 3\begin {pmatrix} -4\\ 8\\ \end {pmatrix}}{4 - 3}$$

or, $$\vec {AB}$$ = $$\frac {\begin {pmatrix} 12\\ 28\\ \end {pmatrix} - \begin {pmatrix} -12\\ 24\\ \end {pmatrix}}1$$

or, $$\vec {AB}$$ = $$\begin {pmatrix} 12 + 12\\ 28 - 24\\ \end {pmatrix}$$

or, $$\vec {AB}$$ = $$\begin {pmatrix} 24\\ 4\\ \end {pmatrix}$$

∴ $$\vec {AB}$$ = 24$$\vec i$$ + 4$$\vec j$$ Ans

Here,

$$\vec {OA}$$ = $$\begin {pmatrix} \sqrt 3\\ 1\\ \end {pmatrix}$$ and $$\vec {OB}$$ = $$\begin {pmatrix} \sqrt 3\\ 3\sqrt 3\\ \end {pmatrix}$$

Let: x1 = $$\sqrt 3$$, y1 = 1, x2 = $$\sqrt 3$$ and y2 = 3$$\sqrt 3$$

Let: $$\theta$$ be the angle between $$\vec {OA}$$ and $$\vec {OB}$$

cos$$\theta$$ = $$\frac {x_1x_2 + y_1y_2}{\sqrt {x_1^2 + y_1^2} \sqrt {x_2^2 + y_2^2}}$$

or, cos $$\angle$$AOB = $$\frac {\sqrt 3 × \sqrt 3 + 1 × 3\sqrt 3}{\sqrt {(\sqrt 3)^2 + 1^2} \sqrt {(\sqrt 3)^2 + (3\sqrt 3)^2}}$$

or, cos $$\angle$$AOB = $$\frac {3 + 3\sqrt 3}{\sqrt {3 + 1}. \sqrt {3 + 27}}$$

or, cos $$\angle$$AOB = $$\frac {3 + 3\sqrt 3}{2\sqrt {30}}$$

or, cos $$\angle$$AOB = $$\frac {8.197}{10.95}$$

or, cos $$\angle$$AOB = 0.75

or, $$\angle$$AOB = cos-1 (0.75)

∴$$\angle$$AOB = 41.58° Ans

If the vector is $$\vec u$$ and $$\vec v$$ are represented by the two adjacent sides $$\vec {AB}$$ and $$\vec {AD}$$ of a parallelogram ABCD, then the sum $$\vec u$$ + $$\vec v$$ is represented by the diagonal $$\vec {AC}$$.

$$\vec {AB}$$ + $$\vec {AD}$$ = $$\vec {AC}$$

$$\vec u$$ + $$\vec v$$ = $$\vec {AC}$$

Unit vector: A vector whose modulus is 1 is called a unit vector.

$$\vec a$$ = ($$\frac 1{\sqrt 2}$$, $$\frac 1{\sqrt 2}$$) is unit vector.

$$\vec a$$ . $$\vec b$$

= (4$$\vec i$$ - 6$$\vec j$$) . (3$$\vec i$$ + 2$$\vec j$$)

= 12$$\vec {i^2}$$ + 8$$\vec i$$ $$\vec j$$ - 18$$\vec i$$ $$\vec j$$ - 12$$\vec {j^2}$$

= 12× 1 + 8× 0 - 18× 0 - 12× 1 [$$\vec {i^2}$$ = $$\vec {j^2}$$ = 1, $$\vec i$$ . $$\vec j$$ = 0]

= 12 - 12

= 0

$$\vec a$$ . $$\vec b$$ = 0

∴ $$\vec a$$ . $$\vec b$$ are perpendicular each other. Proved

Here,

$$\vec a$$ = (4$$\vec i$$ + 5$$\vec j$$) and $$\vec b$$ = (5$$\vec i$$ - 4$$\vec j$$)

$$\vec a$$ = (4, 5) and $$\vec b$$ = (5, -4)

If $$\theta$$ be the angle between $$\vec a$$ and $$\vec b$$ then:

cos$$\theta$$ = $$\frac {\vec a . \vec b}{\begin {vmatrix} \vec a\\ \end {vmatrix} \begin {vmatrix} \vec b\\ \end {vmatrix}}$$

or, cos$$\theta$$ = $$\frac {(4, 5) . (5, -4)}{\sqrt {4^2 + 5^2} \sqrt {5^2 + (-4)^2}}$$

or, cos$$\theta$$ = $$\frac {20 - 20}{\sqrt {16 + 25} \sqrt {25 + 16}}$$

or, cos$$\theta$$ = $$\frac 0{25 + 16}$$

or, cos$$\theta$$ = 0

or, $$\theta$$ = cos-1 (0)

∴ $$\theta$$ = 90° Ans

Here,

$$\vec a$$ = 10$$\vec i$$ - 7$$\vec j$$ and $$\vec b$$ = 7$$\vec i$$ + 10$$\vec j$$

If $$\theta$$ be the angle between the vectors $$\vec a$$ and $$\vec b$$.

cos$$\theta$$ = $$\frac {\vec a . \vec b}{\begin {vmatrix} \vec a\\ \end {vmatrix} \begin {vmatrix} \vec b\\ \end {vmatrix}}$$

or, cos$$\theta$$ = $$\frac {x_1x_2 + y_1y_2}{\sqrt {x_1^2 + y_1^2} \sqrt {x_2^2 + y_2^2}}$$

or, cos$$\theta$$ = $$\frac {10 × 7 + 10 × (-7)}{\sqrt {(10)^2 + (-7)^2} \sqrt {7^2 + {10}^2}}$$

or, cos$$\theta$$ = $$\frac {70 - 70}{\sqrt {149} \sqrt {149}}$$

or, cos$$\theta$$ = 0°

∴ $$\theta$$ = cos-1(0°) = 90° Ans

In the $$\triangle$$ABC,

Let O be the origin.

Position vector of the point A is = $$\vec {OA}$$ = (-1, -1)

Position vector of the point B is = $$\vec {OB}$$ = (5, -1)

Position vector of the point C is = $$\vec {OC}$$ = (2, 5)

If G be the centroid of the triangle,

= $$\frac 13$$ ($$\vec {OA}$$ + $$\vec {OB}$$ + $$\vec {OC}$$)

= $$\frac 13$$ (- 1 - 1 + 5 - 1 + 2 + 5)

= $$\frac 13$$ (6 + 3)

= 2 + 1

Hence, the position vector of the centroid is (2, 1). Ans

Here,

$$\vec a$$ = 6$$\vec i$$ - 8$$\vec j$$ and $$\vec b$$ = 4$$\vec i$$ + 3$$\vec j$$

$$\vec a$$ . $$\vec b$$

= (6$$\vec i$$ - 8$$\vec j$$) (4$$\vec i$$ + 3$$\vec j$$)

= 6 . 4$$\vec {i^2}$$ + 18$$\vec i$$ $$\vec j$$ - 32 $$\vec i$$ $$\vec j$$ - 24$$\vec {j^2}$$

= 24 . 1 + 18 . 0 - 32 . 0 * 24 . 1 [$$\because$$ $$\vec {i^2}$$ = $$\vec {j^2}$$ = 1 , $$\vec j$$ = $$\vec i$$ $$\vec j$$ = 0]

= 24 - 24

= 0

∴ The dot product of two vector a and b is equal to zero so these are perpendicular to each other. Proved

Here,

$$\vec {OA}$$ = 7$$\vec i$$ - 5$$\vec j$$ and $$\vec {OB}$$ = 5$$\vec i$$ - 7$$\vec j$$

$$\vec a$$ . $$\vec b$$ = x1x2 + y1y2 = 7× 5 + (-5) (-7) = 35 + 35 = 70

$$\begin {vmatrix} \vec a \end {vmatrix}$$ = $$\sqrt {x_1^2 + y_1^2}$$ = $$\sqrt {7^2 + (-5)^2}$$ = $$\sqrt {49 + 25}$$ = $$\sqrt {74}$$

$$\begin {vmatrix} \vec b\end {vmatrix}$$ = $$\sqrt {x_1^2 + y_1^2}$$ = $$\sqrt {5^2 + (-7)^2}$$ = $$\sqrt {25 + 49}$$ = $$\sqrt {74}$$

cos$$\theta$$ = $$\frac {\vec a . \vec b}{\begin {vmatrix} \vec a \end {vmatrix} \begin {vmatrix} \vec b\end {vmatrix}}$$

or, cos$$\theta$$ = $$\frac {70}{\sqrt {74} . \sqrt {74}}$$

or, cos$$\theta$$ = $$\frac {70}{74}$$

or, cos$$\theta$$ = 0.95

or, $$\theta$$ = cos-1 (0.95)

∴ $$\theta$$ = 18.93°

∴ $$\angle$$AOB = 18.93° Ans

Unit Vector: If the magnitude of the vector is 1 such type of vector is known as unit vector.

i.e., $$\vec {OP}$$ = $$\begin {pmatrix} 1\\ 0\\ \end {pmatrix}$$

Let, $$\vec a$$ = 3$$\vec i$$ + 4$$\vec j$$ and $$\vec b$$ = 7$$\vec i$$ + 8$$\vec j$$ are the position vectors of A and B,

Position vector of mid-point $$\vec M$$ = $$\frac 12$$ ($$\vec a$$ + $$\vec b$$)

$$\vec M$$ = $$\frac 12$$ (3$$\vec i$$ + 4$$\vec j$$ + 7$$\vec i$$ + 8$$\vec j$$) = $$\frac 12$$ (10$$\vec i$$ + 12$$\vec j$$) = 5$$\vec i$$ + 6$$\vec j$$ Ans

Here,

$$\vec {OA}$$ = $$\vec a$$, $$\vec {OB}$$ = $$\vec b$$ and$$\vec {AC}$$ = 3$$\vec {AB}$$ then, $$\vec {OC}$$ = ?

Using triangle law of vector addition:

$$\vec {OA}$$ + $$\vec {AB}$$ $$\vec {OB}$$

or, $$\vec {AB}$$ = $$\vec {OB}$$ - $$\vec {OA}$$ = $$\vec b$$ - $$\vec a$$

We know,

$$\vec {AC}$$ = 3$$\vec {AB}$$ = 3($$\vec b$$ - $$\vec a$$) = 3$$\vec b$$ - 3$$\vec a$$

Again,

$$\vec {OA}$$ + $$\vec {AC}$$ = $$\vec {OC}$$

or, $$\vec {OC}$$ = $$\vec a$$ + 3$$\vec b$$ - 3$$\vec a$$

∴ $$\vec {OC}$$ = 3$$\vec b$$ - 2$$\vec a$$ Ans

The two vectors are $$\vec a$$ and $$\vec b$$ are parallel if $$\vec a$$ = m$$\vec b$$ where m is scalar product.

$$\vec a$$ = 3$$\vec i$$ + 4$$\vec j$$ and $$\vec b$$ = 8$$\vec i$$ - 6$$\vec j$$

$$\vec a$$.$$\vec b$$ = x1x2 + y1y2 = 3× 8 + (-4) × 6= 24 - 24 = 0

$$\begin {vmatrix} \vec a\\ \end {vmatrix}$$ = $$\sqrt {x_1^2 + y_1^2}$$ = $$\sqrt {(3)^2 + (4)^2}$$ = $$\sqrt {9 + 16}$$ = $$\sqrt {25}$$ = 5

$$\begin {vmatrix} \vec b\\ \end {vmatrix}$$ = $$\sqrt {x_1^2 + y_1^2}$$ = $$\sqrt {(8)^2 + (-6)^2}$$ = $$\sqrt {64 + 36}$$ = $$\sqrt {100}$$ = 10

We know,

cos$$\theta$$ = $$\frac {\vec a . \vec b}{\begin {vmatrix} \vec a\\ \end {vmatrix} \begin {vmatrix} \vec b\\ \end {vmatrix}}$$ = $$\frac 0{5 × 10}$$ = 0

∴ $$\theta$$ = cos-1 (0) = 90° Ans

Let: $$\vec a$$ = $$\vec i$$ - 3$$\vec j$$ and $$\vec b$$ = 2$$\vec i$$ - 5$$\vec j$$ , m:n = 3:1

Position vector of C ($$\vec {OC}$$) = $$\frac {n\vec a + m \vec b}{m + n}$$

$$\vec c$$ = $$\vec {1 (\vec i - 3\vec j)}{3 + 1}$$

$$\vec c$$ = $$\frac {\vec i - 3\vec j + 6\vec i + 15\vec j}{4}$$

$$\vec c$$ = $$\frac {7\vec i + 12\vec j}4$$ Ans

$$\vec {AB}$$ = $$\begin {pmatrix} x_2 - x_1\\ y_2 - y_1\\ \end {pmatrix}$$ =$$\begin {pmatrix} 2 - 1\\ 5 + 3\\ \end {pmatrix}$$ = $$\begin {pmatrix} 1\\ 8\\ \end {pmatrix}$$

$$\begin {vmatrix} \vec {AB}\\ \end {vmatrix}$$ = $$\sqrt {x^2 + y^2}$$ = $$\sqrt {1^2 + 8^2}$$ = $$\sqrt {1 + 64}$$ = $$\sqrt {65}$$ Ans

Here,

Using triangle law of vector addition,

$$\vec {OC}$$ = $$\vec {OA}$$ + $$\vec {AC}$$

or, $$\vec {OC}$$ = $$\vec a$$ + 5$$\vec {AB}$$

or, $$\vec {OC}$$ = $$\vec a$$ + 5 ($$\vec {OB}$$ - $$\vec {OA}$$)

or, $$\vec {OC}$$ = $$\vec a$$ + 5 ($$\vec b$$ - $$\vec a$$)

or, $$\vec {OC}$$ = $$\vec a$$ + 5$$\vec b$$ - 5$$\vec a$$

∴ $$\vec {OC}$$ = 5$$\vec b$$ - 4$$\vec a$$ Ans

Orthogonal Vectors: If two vectors are perpendicular each other then vector are known as orthogonal vector.

$$\vec p$$ . $$\vec q$$ = 3× 2 - 2× 3 = 0

∴ The dot product od $$\vec p$$ and $$\vec q$$ is equal to zero, so these vectors are orthogonal. Proved

Let: $$\vec {OA}$$ = 3$$\vec i$$ - 2$$\vec j$$ and $$\vec {OB}$$ = 1$$\vec i$$ + 8$$\vec j$$

Mid-point of $$\vec {AB}$$

= $$\frac {\vec {OA} + \vec {OB}}2$$

= $$\frac {3\vec i - 2\vec j + 1\vec i + 8\vec j}2$$

= $$\frac {4\vec i + 6\vec j}2$$

= 2$$\vec i$$ + 3$$\vec j$$

∴ Position vector of the mid-point of $$\vec {AB}$$ = 2$$\vec i$$ + 3$$\vec j$$ Ans

Position Vectors: Let P(x, y) be any point on the co-ordinates axes, then (x, y) is called the position vector of the point P with referred to the origin O. OP is a position vector.

Let: $$\vec a$$ = 3$$\vec i$$ + 4$$\vec j$$ and $$\vec b$$ = -$$\vec i$$ + 2$$\vec j$$

$$\vec m$$ = ?

$$\vec m$$ = $$\frac {\vec a + \vec b}2$$ = $$\frac {3\vec i + 4\vec j - \vec i + 2\vec j}2$$ = $$\frac {2\vec i + 6\vec j}2$$ = $$\vec i$$ + 3$$\vec j$$ Ans

Using triangle law of vector addition,

In $$\triangle$$QRS,

$$\vec {QS}$$ = $$\vec {QR}$$ + $$\vec {RS}$$ ....................................(1)

In $$\triangle$$PQS,

$$\vec {QS}$$ = $$\vec {QP}$$ + $$\vec {PS}$$ ...................................(2)

From (1) and (2),

$$\vec {QR}$$ + $$\vec {RS}$$ = $$\vec {QP}$$ + $$\vec {PS}$$

or, $$\vec {QR}$$ + $$\vec {RS}$$ - $$\vec {PS}$$ = $$\vec {QP}$$

∴ $$\vec {QR}$$ + $$\vec {RS}$$ +$$\vec {SP}$$ = $$\vec {QP}$$ Proved

Given,

$$\vec a$$ = $$\vec i$$ + 3$$\vec j$$ and $$\vec b$$ = 2$$\vec i$$ + $$\vec j$$

Let an anglebetween $$\vec a$$ and $$\vec b$$ be $$\theta$$

So,

cos$$\theta$$ = $$\frac {\vec a . \vec b}{\begin {vmatrix} \vec a \end {vmatrix} \begin {vmatrix} \vec b \end {vmatrix}}$$

or, cos$$\theta$$ = $$\frac {x_1x_2 + y_1y_2}{\sqrt {{x_1}^2 +{y_1}^2} \sqrt {{x_2}^2 + {y_2}^2}}$$ [where x1 = 1, y1 = 3, x2 = 2 and y2 = 1]

or, cos$$\theta$$ = $$\frac {1 × 2 + 3 × 1}{\sqrt {1^2 + 3^2} \sqrt {2^2 + 1^2}}$$

or, cos$$\theta$$ = $$\frac {2 + 3}{\sqrt {10} \sqrt 5}$$

or, cos$$\theta$$ = $$\frac 5{\sqrt {50}}$$

or, cos$$\theta$$ = $$\frac 5{5\sqrt 2}$$

or, cos$$\theta$$ = $$\frac 1{\sqrt 2}$$

or, cos$$\theta$$ = cos 45°

∴ $$\theta$$ = 45° Ans

Null Vector: If the magnitude of vector is equal to zero such type of vector is known as zero or null vector. $$\vec {AA}$$ = 0

If $$\vec a$$ and $$\vec b$$ are perpendicular then,

$$\vec a$$ . $$\vec b) = 0 or, (-5, 3) (p, p+2) = 0 or, -5× p + 3× (p + 2) = 0 or, -5p + 3p + 6 = 0 or, -2p = - 6 or, p = \(\frac 62$$

∴ p = 3 Ans

Joining point P and R,

In $$\triangle$$RSP,

$$\vec {RP}$$ = $$\vec {RS}$$ + $$\vec {SP}$$.................................(1)

In $$\triangle$$PQR,

$$\vec {PR}$$ = $$\vec {PQ}$$ + $$\vec {QR}$$..............................(2)

Adding equation (1) and (2),

$$\vec {RP}$$ + $$\vec {PR}$$ = $$\vec {RS}$$ + $$\vec {SP}$$ + $$\vec {PQ}$$ + $$\vec {QR}$$

or, -$$\vec {PR}$$+ $$\vec {PR}$$ = $$\vec {RS}$$ + $$\vec {SP}$$ + $$\vec {PQ}$$ + $$\vec {QR}$$ [$$\because$$ $$\vec {RP}$$ = -$$\vec {PR}$$]

or, 0= $$\vec {RS}$$ + $$\vec {SP}$$ + $$\vec {PQ}$$ + $$\vec {QR}$$

∴ $$\vec {RS}$$ + $$\vec {SP}$$ + $$\vec {PQ}$$ + $$\vec {QR}$$ = 0 Proved

$$\vec {AB}$$, $$\vec {BC}$$ and $$\vec {AC}$$ are the vertices which represent the side of $$\triangle$$ABC using triangle law of addition.

$$\vec {BC}$$ + $$\vec {CA}$$ = $$\vec {BA}$$

or, $$\vec {BC}$$ + $$\vec {CA}$$ = -$$\vec {AB}$$ [$$\because$$ $$\vec {BA}$$ = - $$\vec {AB}$$]

∴ $$\vec {AB}$$ + $$\vec {BC}$$ + $$\vec {CA}$$ = 0 Proved

In $$\triangle$$ABC,

$$\vec {AB}$$ + $$\vec {BC}$$ = $$\vec {AC}$$............................(1)

In $$\triangle$$ACD,

$$\vec {AC}$$ + $$\vec {CD}$$ = $$\vec {AD}$$...........................(2)

In $$\triangle$$ADE,

$$\vec {AD}$$ + $$\vec {DE}$$ = $$\vec {AE}$$.............................(3)

or, $$\vec {AC}$$ + $$\vec {CD}$$ + $$\vec {DE}$$ = $$\vec {AE}$$

or, $$\vec {AB}$$ + $$\vec {BC}$$ + $$\vec {CD}$$ + $$\vec {DE}$$ = $$\vec {AE}$$

∴$$\vec {AB}$$ + $$\vec {BC}$$ + $$\vec {CD}$$ + $$\vec {DE}$$ + $$\vec {EA}$$ = 0 Proved

Here,

Using triangle law of vector addition,

In $$\triangle$$OBC,

$$\vec {BC}$$ = $$\vec {BO}$$ + $$\vec {OC}$$

∴ $$\vec {BC}$$ = -$$\vec b$$ + $$\vec c$$

In $$\triangle$$OAD,

$$\vec {OD}$$ = $$\vec {OA}$$ + $$\vec {AD}$$ = $$\vec {OA}$$ + $$\vec {BC}$$ [$$\vec {AD}$$ = $$\vec {BC}$$]

∴ $$\vec {OD}$$ = $$\vec a$$ - $$\vec b$$ + $$\vec c$$ Ans

0%

12

14

13

15

1

-1

3

2

1

-1

-2

2

4

3

2

5

• ### If (overrightarrow a) and (overrightarrow b) are perpendicular to each other, find the value of m:(overrightarrow a) = -4(overrightarrow i) + 7(overrightarrow j) and  (overrightarrow b) = 14(overrightarrow i) - 3m(overrightarrow j)

8

(-frac{8}{3})

2

6

600

400

450

300

11

10

13

12

• ### The position vectors of points A and B of a line are (egin{pmatrix}1\3\ end{pmatrix}) and (egin{pmatrix}3\5\ end{pmatrix}) respectively.Find the position vector of mid point M of AB.

(egin{pmatrix}5\4\ end{pmatrix})

(egin{pmatrix}1\4\ end{pmatrix})

(egin{pmatrix}2\5\ end{pmatrix})

(egin{pmatrix}2\4\ end{pmatrix})

• ### If the position vectors of the points A and B are 3(overrightarrow i) + 4(overrightarrow j) and 5(overrightarrow i) - 2(overrightarrow j) respectively. Find the position vector of the mid-point M of AB.

2(overrightarrow i) + 3(overrightarrow j)

5(overrightarrow i) + (overrightarrow j)

3(overrightarrow i) + (overrightarrow j)

4(overrightarrow i) + (overrightarrow j)

• ### The position vector of P and Q are 2(overrightarrow i) + 7(overrightarrow j) and 4(overrightarrow i) - 3(overrightarrow j). Find the position vector of a point which divides PQ externally in the ratio of 2:3.

27(overrightarrow j) - 2(overrightarrow i)

25(overrightarrow j) - (overrightarrow i)

23(overrightarrow j) - 2(overrightarrow i)

29(overrightarrow j) - (overrightarrow i)

• ### If the points X(-1, -1), Y(5, 1) and Z(2, 6) are the vertices of triangle XYZ, find the position vector of its centriod.

(egin{pmatrix}2\1\ end{pmatrix})

(egin{pmatrix}1\2\ end{pmatrix})

(egin{pmatrix}2\2\ end{pmatrix})

(egin{pmatrix}2\3\ end{pmatrix})

• ### Find the position vector of a point in the x-axis which divides the line joining the points (2, -1) and (8, 2) in the ratio 1:2.

(egin{pmatrix}2\2\ end{pmatrix})

(egin{pmatrix}4\0\ end{pmatrix})

(egin{pmatrix}4\2\ end{pmatrix})

(egin{pmatrix}2\4\ end{pmatrix})

13

12

14

15

-2

-4

-3

-5

zero

five

eight

two

## DISCUSSIONS ABOUT THIS NOTE

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Forum Time Replies Report

If vector a

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Ask any queries on this note.Altitude of isoscels

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##### Dhananjay Prasad Shah

Prove by vector method that perpendicular drown from vertices to opposite side of triangle are concurrent

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##### bishnu

sabai note lai apps banayara market ma lyaunus..... suvkamana xa.

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